Metric properties

2010015810

Level:

The picture shows a square pyramid. The side of a base square is

a = 10 cm

and the height of the pyramid is

v = 10 cm

. Find the angle

φ

between the lateral edge and the edge of the base of the pyramid.

tg φ = \sqrt{5} ⟹ φ ≐ 65^{\circ} 54^{'}

tg φ = \frac{\sqrt{5}}{5} ⟹ φ ≐ 24^{\circ} 6^{'}

tg \frac{φ}{2} = \frac{\sqrt{5}}{5} ⟹ φ ≐ 48^{\circ} 11^{'}

tg φ = \frac{\sqrt{10}}{2} ⟹ φ ≐ 57^{\circ} 41^{'}

2010015809

Level:

The picture shows a square pyramid

A B C D V

. The side of a base square is

a = 6 cm

and the height of the pyramid is

v = 8 cm

. Find the angle

φ

between the opposite lateral edges (the angle

A V C

tg \frac{φ}{2} = \frac{3 \sqrt{2}}{8} ⟹ φ ≐ 55^{\circ} 53^{'}

tg φ = \frac{3 \sqrt{2}}{8} ⟹ φ ≐ 27^{\circ} 56^{'}

tg \frac{φ}{2} = \frac{3}{8} ⟹ φ ≐ 41^{\circ} 7^{'}

tg \frac{φ}{2} = \frac{8}{3 \sqrt{2}} ⟹ φ ≐ 124^{\circ} 7^{'}

2010015808

Level:

The picture shows a square pyramid. The side of a base square is

a = 6 cm

and the height of the pyramid is

v = 10 cm

. Find the angle

φ

tg φ = \frac{10}{3 \sqrt{2}} ⟹ φ ≐ 67^{\circ}

tg φ = \frac{10}{3} ⟹ φ ≐ 73^{\circ} 18^{'}

tg \frac{φ}{2} = \frac{3 \sqrt{2}}{10} ⟹ φ ≐ 45^{\circ} 59^{'}

tg \frac{φ}{2} = \frac{3}{10} ⟹ φ ≐ 33^{\circ} 24^{'}

2010015807

Level:

The sides of a rectangular box shown in the picture are

a = 3 cm

b = 4 cm

, and

c = 12 cm

. The space diagonal is

u_{t}

and the shortest face diagonal is

u_{s}

. Find the ratio

u_{s} : u_{t}

5 : 13

13 : 5

13 \sqrt{10} : 40

4 \sqrt{10} : 13

2010015806

Level:

The side of a regular hexagonal prism

A B C D E F A^{'} B^{'} C^{'} D^{'} E^{'} F^{'}

shown in the picture is

a = 3 cm

and the height is

v = 8 cm

. Find the angle between the diagonal

A C^{'}

and the base plane

A B C

(round your result to the nearest degree).

57^{\circ}

53^{\circ}

33^{\circ}

38^{\circ}

2010015805

Level:

A cuboid has sides

a = 6 cm

and

b = 8 cm

, and the space diagonal

u = 11 cm

. Find the length of the side

c

(see the picture).

\sqrt{21} cm

\sqrt{221} cm

21 cm

10 cm

2010015804

Level:

The base

A B C D

of a square pyramid

A B C D V

has an edge of

6 cm

. The height of the pyramid is

3 \sqrt{2} cm

. Find the distance between the point

A

and the line

C V

(see the picture).

6 cm

3 \sqrt{3} cm

9 cm

3 \sqrt{2} cm

2010015803

Level:

Let

A B C D

be a regular tetrahedron with the height of

3 \sqrt{6} cm

. Find the edge length of the tetrahedron (see the picture).

9 cm

9 \sqrt{2} cm

6 \sqrt{2} cm

3 \sqrt{6} cm

2010015802

Level:

Let

A B C D E F V

be a regular hexagonal pyramid with a base edge length of

4 cm

and a height of

8 cm

. Find the distance between the point

V

and the line

B D

(see the picture).

2 \sqrt{17} cm

4 \sqrt{3} cm

2 \sqrt{19} cm

2 \sqrt{20} cm

2010015801

Level:

Let

A B C D E F A^{'} B^{'} C^{'} D^{'} E^{'} F^{'}

be a regular hexagonal prism with the base edge length of

4 cm

and the height of

6 cm

. Find the distance between the lines

F A

and

D^{'} C^{'}

(see the picture).

2 \sqrt{21} cm

4 \sqrt{3} cm

10 cm

2 \sqrt{13} cm

2010015810

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